Tuesday, 1 May 2007

Numbers have been represented using many different notations over the years. We use what is called the "Hindu-Arabic" system, but there have been many others in the past. There are systems developed by the Romans, Babylonians, Mayans, Egyptians, and many others.

It's a part of the standard elementary school syllabus here in New South Wales to teach about different number systems. This is a fun topic for including a bit of history in a class; and thinking outside the box, and helping students distinguish between the abstract concept of a number and the symbols we use to denote it.

One common feature of such lessons is to show the advantages of "our" Hindu-Arabic system over all those others, especially for manual calculations. But wouldn't it be fun if we could extend the lesson with a number system that was better than what we are using now?

Wait no longer. I now reveal for the world a new, improved number system!

The idea is simple. It is a minor modification to the conventional decimal system. It is still in base 10, so conversion between old fashioned notation and the new improved notation is easy.

Instead of using digits from 0 to 9, the new system uses digits from -4 to 5.

The symbols adopted for each digit are not all that important. One approach would be to add a sign designation of some kind to the digit, such as a strike through line. In this post I'll go with something easy to type, and to remember. Digits 0 to 5 are represented as normal. The four negative digits are represented using the first four letters of the Greek alphabet.

Negative digits

α

β

γ

δ

Value

-1

-2

-3

-4

A sequence of digits denotes a number just as in the Hindu-Arabic system. There is the ones place, the tens place, and so on. For example, 15 is unchanged in this new system; but 16 becomes 2δ in the new. This is 2 tens, and -4 ones, and so denotes the same number.

The speed of light is 299792468 m/s in old notation. In the new notation, it is 300βα25γβ m/s.

Why would anyone do this? It takes time to get used to a new number system, but if you grew up with it, you'd see a number of significant advantages that make it arguably superior to the system we are currently using.

Adding up large columns is easier

When you sum a column of n digits, randomly selected, the old system has a distribution of results with mean 4.5n and standard deviation around about 3*sqrt(n). In the new system, the standard deviation is unchanged, but the sum has mean 0.5n; about 9 times smaller. For a column of ten digits, the sum is, on average, around 5, with a standard deviation of around 10. You often don't carry anything at all; when you do carry any overflow to the next column, the amount carried is small.

Memorizing the times tables is easier

You now only need to remember up to your 5 times table. Multiplying digits of opposite sign just means reversing the sign of all digits in the answer. That's not quite true; but it will be true if we add the digit ε for -5. With ε, representations are no longer unique, but removing such a digit is easy. Just replace with 5, and carry α (that is, -1) to the next digit. Eg: 2ε becomes 15, βε becomes γ5.

Here is a complete multiplication table.

ε

δ

γ

β

α

0

1

2

3

4

5

ε

25

20

15

10

5

0

α5

α0

β5

β0

γ5

δ

20

2δ

12

1β

4

0

δ

α2

αβ

β4

β0

γ

15

12

1α

1δ

3

0

γ

α4

α1

αβ

β5

β

10

1β

1δ

4

2

0

β

δ

α4

α2

α0

α

5

4

3

2

1

0

α

β

γ

δ

ε

0

0

0

0

0

0

0

0

0

0

0

0

1

α5

δ

γ

β

α

0

1

2

3

4

5

2

α0

α2

α4

δ

β

0

2

4

1δ

1β

10

3

β5

αβ

α1

α4

γ

0

3

1δ

1α

12

15

4

β0

β4

αβ

α2

δ

0

4

1β

12

2δ

20

5

γ5

γ0

β5

β0

α5

0

5

10

15

20

25

As a persuasive demonstration of the power of this system, we can calculate 257*473. In new notation, this is 3δγ*5γ3. Here are the multiplication tableaux, side by side:

In the first tableau, there is only one instance in which the sum requires a digit to be carried; in the second tableau you must carry three times; and one of those requires a two to be carried. This is typical of such calculations.

Long division is more robust

As with multiplication, the simplified times tables makes long division easier to carry through. If we are willing to be a bit flexible about digits, there are further advantages.

Have you ever been working through a long division problem, and made the wrong guess for the next digit? For example, consider division of 2930872 by 7364. In the current cumbersome system, you know that the answer is close to 400, so you have to guess the first digit as either 3 or 4. If you pick 4, then you'll need to back up the calculation, as the answer is actually 398.

In this new notation, getting the wrong digit by one is not a big problem, as long as you are willing to extend your repertoire to include some additional digits, like 6 and 7, or ε and ζ. This also means that you can make much cruder guesses for each digit in the division tableau, confident that recovery will be easy.

For example, here is a long division tableau for 121561 divided by 473. The answer, by the way, is 257, which in new notation makes finding the first digit a bit harder.

3δγ --------5γ3 ) 122δδ1 142α ---- β0γδ β11β ---- αδβ1 αδβ1 ---- 0

But suppose that you got that first digit wrong, and proposed a 2 instead. It is a perfectly good strategy in this notation to ignore all but the most significant digit, and in this case we can guess a 2 because 5 goes into 12 about twice.

2 --------5γ3 ) 122δδ1 1α5δ ---- 3γ0δ

At this point, you can guess a 5 or a 6 for the next digit, because 3γ is 27, and 5 goes into 27 a bit over 5 times. We'll go with 6, because the next digit in the divisor 5γ3 is negative, so overestimates are sensible. Note that we are using a "non-standard" digit here.

Of course, we need to "normalize" the result, by converting all non-standard digits, but this is a simple operation. The 6 is replaced by 1δ, and the 1 is carried over the next place, so the final result is, as before, 3δγ. This is 257, in positive digits.

This example gives the game away somewhat. We can regard the "new" system as allowing for alternative ways to write numbers in our existing system, by introducing some non-standard digits.

Negative numbers are easy

Negative numbers are nothing special in this system. They just use a negative valued digit in the most significant place.

Rounding is easier

If you take a fixed number of digits from the start of a number, you are effectively rounding. Actually, this is not quite true, and a deeper consideration of this issue suggests that the new system is actually more difficult for rounding to a nearest digit. So I'll gloss over the point and skip quickly to the conclusion.

Conclusion

I don't actually expect this new system to sweep the world. With some practice, it does help with certain kinds of manual calculations. But is it really worth the pain of learning a new system?

The real utility of the system is that it may help students think outside the box a bit with respect to number systems. It might be useful to have students imagine being aliens from a planet where negative digits are used from earliest childhood, and then try to think through what kinds of consequences might follow in manipulating numbers. Unfortunately, long division seems to be a lost art, so the force of this demonstration is somewhat diluted. The multiplication tableaux remain an intriguing exercise.

Update. This post is part of the carnival of mathematics. Check out what other math bloggers are doing at Carnival of Mathematics Edition #7, hosted by nonoscience.Update, June 3, 18:05. This system is probably one that has occurred to many people. The earliest published reference I have found is Lehmer, D.N. (1903) in Note on Negative Digits, Science, Vol. 17, No. 430. (Mar. 27, 1903), p. 514. (Online at JSTOR). It would not surprise me to find older references still.

14 comments:

What I like the most is that it's easy to transform from our usual positive-digit-base-10 to negative digits and back again. I do on-paper arithmetic a lot, and I think I'll be using this now, though I'll add (-5) at least, and use positive digits for normal forms.

Hail and well met! It's a real pleasure to hear from you again! I've not been on arc now for years.

I'm also delighted that someone else gets it with this number system. I've cheated a bit calling it a new number system. It works best as a convenient extension to the existing number system. In different contexts there can be advantages in manual calculation methods to using any of the digits from -9 to 9; or even beyond in principle.

As a party trick in those limited contexts where the trick is actually appreciated, you need to be able to express the numbers vebally. I just put a "b" (buh) sound at the front of any negated digit. For example:

I haven't been seen on arc for ages either. I don't think they've found a new moderator.

The advantage that I can see of using the full range -9 to 9 means that the existing investment in multiplication tables isn't wasted.

I just did some doodlings, and I can report that the usual pencil-and-paper method for computing square roots is also much better in this system, thanks to the self-adjusting nature, much like with long division.

I agree with the carry-over part making the job much easier. But you might be cheating with division. To compare apples with apples, you ought to add alpha, beta,... to the original decimal system as well. That way, even if you guessed incorrect, it would be fine because a number equivalent of 10 or 11... (or -1, -2,...) would come next... just as in your new system.

In fact, some of the "vedic math" tricks, developed close to a millenia and half ago, allows one to do something similar to compute difficult problems rather quickly.

Niket, you are completely correct! The flexibility in division is basically a tolerance of additional digits. I recognize this point in the second comment. Using digits for ten, eleven and twelve gives you the same capacity in decimal for continuing the tableau after having chosen the earlier digit one too small.

I recently read the article from Duo Quartuncia about the "New Improved Number System" and find that it interesting while trying to get a good feel for some of the properties that this improvement enhances to the base-10 number system.

It would be good for you to post more simple examples of base operations to illuminate the various operation in comparrison to the base-10 systems.

From what I have read, I like what I have seen very much and it may ( or may not in the worse case) bring light to some research that I am doing in being able to look at number systems in a different way.

I would also be very interested in reading any additional papers on this subjust that may be about as well as eagerly looking forward to your up and coming post relating to higher operations like "squaring" and higher powers as well as the "roots".

Additionally, if you (Duo) or any one else on the blog has some more information (and examples) on this very interesting topic then I would really like to hear from you since I am stiull trying to get use to the system.

I am working to get a good feel for the "extended" Base-10 number system as I think that it exhibits some nice properties that are of great benefit.

currently, I have a feel for conversion of numbers B10 to EB10 and back, but do not have a complete feel for some of the things that happen in addition and subtractions.

For Example:

alpha + beta = gammaalpha - beta = 1

but,

gamma - delta = 1 (????)gamma + delta = ????

and is,

5 + 1 = alpha or ( 1 alpha )

Additionally, on the multiplication table you have included (eplison) which brings in symmetry but in your division problem, you are not using it and only going from (-4 to 5).

could you please explain a little more about this and how using additional tokens enhances the "tolerance" in such operations as division.

In the division problem presented, I can see how things progress although I am still working on understand the addition operations which should come in time, and you mentioned that finding the first digit is harder, but that we can basically ignore all but the most significant digit at a time.

I am wondering what would have happened if you tried to divide this instead:

122δδ1 / 3δγ

and started off by choosing a 4 since 3*4 = 12,

122δδ1 / 3δγ = 4??

then would the system still be "self-correcting" as I really like that property and would like to understand it better in t his system.

Sorry for the long post but I find this system to be very exciting and think that I will be able to use it a lot in the future.

Hey Lonnie; welcome to my corner of the web. Thanks for the interest you've shown in this!

You have the calculations with alpha and beta and gamma and delta correct.

The extra one you ask about is gamma plus delta. The answer is:γ + δ = α3

This is analogous to 3 + 4, except with negative digits. 3 + 4 is seven, which is ten minus three. Reverse it for the negatives, and you get negative 10 plus three, or α3.

5 + 1 is 1δ (ten minus four).

I did include ε in the multiplication table. It's not strictly necessary, but was nice to show the symmetries.

The long division allows "tolerance" as long as we can use a few non-standard digits. That is, if you guess a digit incorrectly, then you can compensate in the next digit; but only by using something non-standard; such as 6 or more, or ε or less. As niket points out, you can also do this with conventional numbers, by extending 0 to 9 with a couple of small negative digits, and introducing digits with values ten, eleven and twelve.

After you obtain an answer, you may want to remove the non-standard digits by normalizing.

You ask about 122δδ1 / 3δγ and propose 4 as the first digit. I can't show easily how it would work in comments, as I don't have html tags to line it up nicely. But here's an attempt.

Pick 4 as the first digit.

Then 4 * 3δγ = 103β

Subtract 122δ - 103β = 2αβ

Carry down the next digit, and you have a new division 2αβδ1 / 3δγ

Now the next digit has to be pretty big, and this is where tolerance comes in. We can pick a non-standard digit, large enough to cover the shortfall in our previous guess.

3 goes into 2α more than 6 times; and since the next digit is after the 3 is negative, we probably want something more than 6 even. Pick 7. Of course, you are going to need to remember the times table for any additional non-standard digits. Fortunately I do remember my seven times table, so here we go...

Multiply 3δγ by 7 to have 2β0αSubtract 2αβδ - 2β0α = 1βγCarry the final digit, and then the next division step is 1βγ / 3δγ

The next digit is 3, and we have the answer now with only positive digits!

The final tableau looks like this, using underscores to help with spacing.

A delayed comment, but you may be pleased to know that this system has been around for a long time. I first saw it in a New Scientist article back in the late 70's. And if you look up "balanced ternary" notation, you'll discover it is the same system but in base 3, using the digits "-1", "0" and "1". Knuth uses it a lot.

Wikipedia has balanced ternary http://en.wikipedia.org/wiki/Balanced_ternary and http://en.wikipedia.org/wiki/Negabinary "Negative numerical bases were first considered by Vittorio Grunwald in his work Giornale di Matematiche di Battaglini, published in 1885. Grunwald gave algorithms for performing addition, subtraction, multiplication, division, root extraction, divisibility tests, and radix conversion. Negative bases were later independently rediscovered by A. J. Kempner in 1936 and Z. Pawlak and A. Wakulicz in 1959.

Negabinary was first implemented in computer hardware in the experimental Polish computers SKRZAT 1 and BINEG in 1950. Implementations since then have been rare." [Some references included.] teal4two

Duae Quartunciae

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I've been active on the internet for many years in various guises. My real name is Chris Ho-Stuart, but I usually leave that for personal life off-line.
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